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Chapter 2: Problem 126

If \(\mathbf{x}\) is a vector such that \([\mathbf{p} \mathbf{q} \mathbf{r}] \mathbf{x}=(\mathbf{p} \times \mathbf{q}) \times \mathbf{r}\), then \(\mathbf{x}\) is \((a) c(\hat{i}-2) \hat{j}+\hat{\mathbf{k}})\) (b) a unit vector (c) indeterminate, as \([\mathbf{p} \mathbf{q} \mathbf{r}](\) d \()-(\hat{\mathbf{i}}-2 \mathbf{j}+\hat{\mathbf{k}}) / 2\)

Short Answer

Expert verified
The vector \( \mathbf{x} \) is indeterminate.

Step by step solution

01

Understand the Problem

We are given a vector equation \[[\mathbf{p} \mathbf{q} \mathbf{r}] \mathbf{x} = (\mathbf{p} \times \mathbf{q}) \times \mathbf{r}\]and need to determine the nature of the vector \( \mathbf{x} \). The goal is to infer whether \( \mathbf{x} \) is equal to any particular expression among the given options.
02

Recall the Vector Triple Product Identity

Apply the vector triple product identity: \[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a}\]Here, substitute \(\mathbf{a} = \mathbf{p}, \mathbf{b} = \mathbf{q}, \mathbf{c} = \mathbf{r}\). The equation becomes:\[(\mathbf{p} \times \mathbf{q}) \times \mathbf{r} = (\mathbf{p} \cdot \mathbf{r})\mathbf{q} - (\mathbf{q} \cdot \mathbf{r})\mathbf{p}\].
03

Substitute and Simplify

Equate both sides of the original equation using the triple product identity:\[[\mathbf{p} \mathbf{q} \mathbf{r}] \mathbf{x} = (\mathbf{p} \cdot \mathbf{r})\mathbf{q} - (\mathbf{q} \cdot \mathbf{r})\mathbf{p}\]This equation implies that \( {\mathbf{x} = \left((\mathbf{p} \cdot \mathbf{r})\mathbf{q} - (\mathbf{q} \cdot \mathbf{r})\mathbf{p} \right)/[\mathbf{p} \mathbf{q} \mathbf{r}]}\)
04

Analyze the Options

Evaluate what kind of vector \( \mathbf{x} \) turns out to be:- The expression obtained does not reduce \( \mathbf{x} \) to a single scalar multiple or simple vector form.- With lack of numeric constraints on \( \mathbf{p}, \mathbf{q}, \mathbf{r} \), \( \mathbf{x} \) remains complex.
05

Conclude the Result

From the evaluation in the previous step, the vector \( \mathbf{x} \) is based on the specific components and arrangements of \( \mathbf{p}, \mathbf{q}, \mathbf{r} \), which makes the vector \( \mathbf{x} \) indeterminate. The correct answer is (c) indeterminate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Triple Product
The vector triple product is an important concept in vector algebra. It's used when taking the cross product of two vectors, \( \mathbf{a} \times \mathbf{b} \), and then crossing that resultant vector with a third vector, \( \mathbf{c} \). The vector triple product identity expresses this in a simpler form. This is given by the formula: \[ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a} \] Key points to remember:
  • The result is expressed as a summation of two scaled vectors, \(\mathbf{b}\) and \(\mathbf{a}\).
  • You utilize both the dot and cross products.
  • The triple product can often simplify complex vector expressions.
This identity is not only a simplifying tool but also crucial for various applications such as calculating torques and studying rotational dynamics.
Vector Equation
A vector equation involves vectors and operations between them, such as addition, subtraction, or the dot and cross products. A specific example from the original exercise is given by the expression: \[[\mathbf{p} \mathbf{q} \mathbf{r}] \mathbf{x} = (\mathbf{p} \times \mathbf{q}) \times \mathbf{r}\] This equation relates the vector \( \mathbf{x} \) to the specific combination of the vectors \( \mathbf{p}, \mathbf{q}, \) and \( \mathbf{r} \). Points to understand:
  • The left side of the equation, \([\mathbf{p} \mathbf{q} \mathbf{r}] \mathbf{x} \), represents a determinant involving \( \mathbf{p}, \mathbf{q}, \mathbf{r}, \) and \( \mathbf{x} \).
  • The right side represents the triple product, simplifying to factors of \( \mathbf{q} \) and \( \mathbf{p} \).
  • Such equations can be used to model various physical systems and make calculations based on vector dynamics.
Understanding vector equations is crucial for solving problems involving forces, velocity, and other vector-based quantities.
Vector Identity
Vector identities are equations that hold true for all vectors and are widely used in simplifying vector calculations. The vector triple product identity is one such identity. They are used to derive useful results in physics and engineering. Here鈥檚 how vector identities help:
  • They provide validated relationships, which reduce computational complexity.
  • They can transform complex vector expressions into simpler forms. For instance, converting a cross product within a cross product into dot products, as shown in the triple product identity.
  • Common identities include the distributive law for vectors, dot product properties, and the rules for working with unit vectors.
Mastering these identities empowers you to tackle vector-related problems with increased efficiency. They form the backbone of vector algebra used in fields like electromagnetism and fluid dynamics.

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Most popular questions from this chapter

If \(a\) and \(b\) are vectors in space given by \(a=\frac{\hat{\mathbf{i}}-2 \hat{\mathbf{j}}}{\sqrt{5}}\) and \(\mathbf{b}=\frac{2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}}{\sqrt{14}}\), then the value of \((2 \mathrm{a}+\mathrm{b}) \cdot[(\mathrm{a} \times \mathrm{b}) \times(\mathrm{a}-2 \mathrm{~b})]\) is \([\) Integer Type Question, 2010]

If the angle between the vectors \(\mathbf{a}=\hat{\mathbf{i}}+(\cos x) \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{b}=\left(\sin ^{2} x-\sin x\right) \hat{\mathbf{i}}-(\cos x) \hat{j}+(3-4 \sin x) \hat{k}\) is obtuse and \(x \in\left(0, \frac{\pi}{2}\right)\), then the exhaustive set of values of \(^{\prime} x^{\prime}\) is equal to (a) \(x \in\left(0, \frac{\pi}{6}\right)\) (b) \(x \in\left(\frac{\pi}{6}, \frac{\pi}{2}\right)\) (c) \(x \in\left(\frac{\pi}{6}, \frac{\pi}{3}\right)\) (d) \(x \in\left(\frac{\pi}{3}, \frac{\pi}{2}\right)\)

Let \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) be unit vectors such that \(\mathrm{a}+\mathrm{b}+\mathrm{c}=0\). Which one of the following is correct? [Single Correct Type, IITJEE 2007] (a) \(a \times b=b \times c=c \times a=0\) (b) \(a \times b=b \times c=c \times a \neq 0\) (c) \(b \times b=b \times c=a \times c=0\) (d) \(\mathbf{a} \times \mathrm{b}, \mathrm{b} \times c, \mathrm{c} \times \mathrm{a}\) are mutually perpendicular

If \(\alpha\) and \(\beta\) are two perpendicular unit vectors such that \(\mathbf{x}=\hat{\boldsymbol{\beta}}-(\alpha \times \mathbf{x})\), then the value of \(4|\mathbf{x}|^{2}\) is.

\(\mathbf{A}, \mathbf{B}\) and \(\mathbf{C}\) are three vectors given by \(2 \hat{\mathbf{i}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}\) Then, find \(\mathbf{R}\), which satisfies the relation \(\mathbf{R} \times \mathbf{B}=\mathbf{C} \times \mathbf{B}\) and \(\mathbf{R} \cdot \mathbf{A}=\mathbf{0}\).
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